Logarithmic calculating device



June 16, 1925.

1 1,541,871 F. o. STILLMAN, I

LOGARITHMIC CALCULATING DEVICE Filed April 17, 1922 a c s [NVf/VTORPatented June 16, 1925.

UNITED STATES PATENT OFFICE.

LOGARITHMIC CALCULATING DEVICE.

Application tiled April 17, 1922. Serial No. 554,348.

To all who-m. it may concern:

Be it known that I, FREDERICK 0. Sum,- MAN, a citizen of the UnitedStates, and a resident of Melrose, in the county of Middlesex andCounnonwealth of Massachusetts,

have invented new and useful Improvements in Logaritlunic CalculatingDevices, of which the following is a specification.

This invention relates to a logarithmic rule type, having one or morelogarithmic scales (numerical values plotted proportional to equallydivided logarithmic distances) divided into two equal linear portionswhich are placed one below theother in registry. These scales togetherwith any auxiliary scales or data are printed on or into an enamel(preferably white) having a celluloid or similar base, which enamel iscoated on wood, metal, paper, or any suitable'substance. The said enameland printing is made water-proof and protected for wear by a coat oftransparent similar enamel applied over the printing and first namedenamel.

The invention in its preferred form is an improvement over the ordinaryMannheim slide-rule in that:

(l) lVhile being just as easily and quickly operated, the former gives,for equal length of rule, four place accuracy instead of three;

(2) The celluloid coating and printing r operation while forming a verydurable surface are much less expensive than engraving in seasonedcelluloid strips.

This invention is further an improvement on divided logarithmic scaleslide-rules for the special case of a slide-rule having each logarithmicscale divided into two equal linear portions placed one below the otherin registry: no extra mechanical operation and practically no mentaloperation are necessary to determine the portion of the logarithmicscale in which the answer is to be found. The cursor of the slide-ruledescribed in this invention is constructed so that no opaquematerial-obscures any parts of the scales from view.

Other objects and advantages of this invention will be pointed out inthe following description and accompanying drawings. The slide-rule typeonly is discussed and il- ,vided scales.

calculating device, preferably of the slider lustratod, but the sameprinciple is applicable to any of the other types of construction for loarithmic calculating devices.

The preterred embodiment of my invention is illustrated by the followingfigures:

Figure 1 shows a top view of the rule and cursor giving logarithmic andequally di- (The application of these equally divided scales isdescribed in Patent No. 1,250,379, granted December 18, 1917 to Henry M.Schleicher and myself).

Figure 2 shows a bottom view of the rule and cursor.

Figure 3 shows an end view of the rule and cursor.

The mechanical features relating to the slide-rule construction compriseupper and lower guide members Gr and G respectively, a slide member S,and a cursor C longitudinally movable on the guide members. lVith theexception of the cursor the mechanical construction is entirely the sameas in the Mannheim, the slide having tongues adapted for reciprocationin grooves inthe guide members, the surfaces being always flush andparallel. The cursor C consists of a channel (preferably of metal), 1;two blocks of fibre or other substance, 2, each provided with grooves,5. The said channel and grooved blocks when assembled restrict movementof the glass, 3, transversely of the slide-rule. The screws or bolts, 4,prevent movement of the glass in the cursor longitudinally of the ruleand secure the blocks to the channel. Spring, 6, holds the cursor in anylongitudinal position of the guides. The hair-line, 7, serves toregister with any longitudinal position on the guide scales.

The remaining discussion is confined to the mathematical featuresrelating to the preferred form of my invention. As shown in Figure 1,the scales on the guide and slide members (with the exception of theequall divided scales, L, L/2, L/3) comprise in eac case two consecutivesuperposed registering portions of a continuous logarithmic scale. Theseportions may be numbered (or otherwise designated by letters or otherindicia). The registering ends of the portions are termed indices. Thesecond portion begins where the first portion leaves off but at theopposite index, Thus the first portion ends 105 and the second portionbegins with 3162.

The equally divided scales described in Patent N 0. 1,250,379 accompanylogarithmic scale portions numbered 0, 1, 2, 3, 4, n1, but it is evidentthat the same rinciple holds no matter how the 10 arit mic portions aredesignated. For suc equally divided scales accompanying continuouslogarithmic scales each comprising two consecutive su erposedregistering portions designated by, say, N and N in consecutive order,the tables .in the aforesaid patent become:

Log portion Ni I-Ni I-Nl I-Ni III-Ni Il-N:

Log portion N: II-Ni Furthermore, the uniform numbering does not need tobe 0 to 10; it can be 0 to any integer or fraction thereof. Otherwisethe description given in the aforesaid patent applies herein.

The preferred arrangement of scales illustrated in Figure 1 is asfollows:

(1) On the upper guide G (a) An equally divided scale marked L(signifying log) numbered from 1 to 5 and subdivided as far asracticable.

(b) An equally divided scale marked I1/2 (signifying log divided by 2)comprising two sections each one-half as long. as scale L and eachdivided and numbered as said scale L. The two sections from left toright are designated by Roman numerals II-N: II-Ni I-Ni III-N1 I and II,respectively.

(a) An equally divided scale marked I l/ifi (signifying log divided by3) comprising tiree sections each one-third as long as scale L and eachdivided and numbered as said scale L. The three sections from left toright are designated by Roman numerals I, II, and III, respectively.

(at) The first portion of a logarithmic scale marked N, (signifyingfirst portion of the number scale) beginning at the left index with 10and ending at the right index with 31.62.

(2) On the slide S- (a) The first portion of a logarithmic scale markedN identically the same as in (1) (d).

(b) The final portion of a cologarithmic scale marked R (signifying thatportion of a reciprocal scale corresponding to N beginning on the rightwith 31.62 and ending on the left with 100.

(c) The first portion of a cologarithmic scale marked R (signifying thatportion of a reciprocal scale corresponding to N beginning on the rightwith 10 and ending on the left with 31.62.

((1) The final portion of a logarithmic scale marked N (signifyingsecond portion of the number scale) beginning on the left with 31.62 andending on the right with 100. i

(3) On the lower guide G (w) The final ortion of a logarithmic scalemarked N identically the same as in 2 d (b) Ir'igonometric functions orany factors may be plotted proportional to their logarithms.

The equally divided scale L represents the logarithms and the so calledlogarithmic scales represent numbers laid ofi'. roportional to theirlogarithms. There ore, multiplication or division may be performed byadding or subtracting, respectivel the distances corresponding to thenum ers. If the use of one slide index brings the desired number settingoff scale (outside the indices of the guide scales), the other slideindex is to be used, since the same reading is obtained as if the scaleswere unbroken and repeated to satisfy any logarithmic characteristic.

The arrangement of scales given above makes it possible;

(1) To register the indices of the slide with any position of eitherportion of the logarithmic scale on the guides without the use of thecursor.

(2) To determine, with the aid of numbers, letters, or other indicia theportion of the logarithmic scale in which the answer of anymultiplication or division operation occurs, by the mechanicaloperations with which the said mathematical operations are performed.Figure 1 shows the digit 1 to the'left of the left slide index and thedigit 2 to the right of the right slide index. The said digit which isat the index that is on scale, after the slide and cursor setting havebeen made, indicates the portion of the logarithmic scale in which theanswer is to be found. If the factors occur in like scale portions, thesub-numerals of N R N,, R, correspond directly with the said digit. Ifthe factors occur in unlike scale portions, the subnumeral 1 correspondswith digit 2 and subnumez'zil 2 corresponds with digit 1. Like scaleportions are two upper scale portions (designated by subnumeral 1, e. g.N or R or two lower scale portions (designated by subnumeral 2, c. g.

N or R The application of this principle will be made clear in thefollowing directions for mathematical operations.

MuZtipZicati0n.One of two methods may be used according to which is moreconvenient.

Method 1. Set the cross-hair of the cursor in registry with the firstfactor on the guide scale portion N or N (whichever contains thefactor). Bring the second factor under the cross-hair using scaleportion R, or R The answer is at the slide index which is on scale inguide scale portion N or N,. If the factors are in N, and R or N and Rthe digit 1 or 2, respectively, indicates that the answer is in the aidescale portion N l or N according to wiether the left or the right indexof the slide is on scale. If the factors are in N and R or N and R thedigit 1 indicates that the answer is in guide scale portion N,, and thedigit 2 indicates that it is in guide scale portion N For example,multiply 2 times 6. Set the cross-hair of cursor to 2 in the upper guidescale )ortion N Bring 6 in R under the cross-hair. The left slide indexis on scale and the factors occur in like scale portion. Therefore, theanswer 12 is at the index in the upper guide scale portion N Method 2.Set one index of the slide to the first factor in the guide scale. Bringthe cross-hair of the cursor to the second factor in slide scale portionN or N (If the second factor is off scale bring the other slide index tothe first factor.) The answer is under the cross-hair in the upper guidescale portion N if the factors occur in like scale portions with theleft slide index on scale or if the factors occur in unlike scaleportions with the right slide index on scale;'

otherwise the answer is in lower portion N For example, multiply 2 times6. Set the right slide index to 2 in the upper guide scale portion NBring the cross-hair of the cursor to 6 in slide scale portion N Theright slide index is on scale and the factors occur in unlike scaleportions. Therefore, the answer 12 is under the crosshair in the upperguide scale portion N Dioz'siom-Set the cross-hair of the cursor to thedividend in guide scale portion N or N Bring the divisor to thecross-hair using slide scale portion N or N The guide scale quotient isin the guide scale portion N or N at the index of the slide which is onscale. If the factors occur in like scales the quotient is in guidescale portion N or N according as the left slide index or the rightslide index is on scale. If the factors occur in unlike scales thequotient is in guide scale portion N or N according as the left or theright slide index is on scale.

For example, divide 12 by 6. Set the crosshair of the cursor to 12 inthe upper guide scale portion N,. Bring 6 in the slide scale portion N.under the cross-hair. The right slide index is on scale and the factorsoccur in unlike scale portions. Therefore the quotient 2 is at the indexin upper guide scale portion N,.

I n'v0l'uti(m.'Ihe general rules for involution are described in PatentNo. 1,250,379; rules for square root and cube root only will be givenhere.

, Square root. Set in the guide scale portion N or N z the cross-lair ofthe cursor to the number of which the square root is sou ht. Take thereadin in the scale L. ote whether the number is in scale portion N or N2 and whether it has the form of m or 10m (which standard mathematicalform is explained in Patent No. 1,250,379). Under these two types ofheadings for square roots in the above table will be found a Romannumeral accompanying N or N,. The Roman numeral I or II designates thesection of scale L/2 in which the L scale reading is to be set. Havingmade this setting the said N or N indicate the guide scale portion inwhich the square root of the number is to be found. Rules applying onlyto the special case embodied in this invention are as follows. If thenumber is in scale portion N,.the L scale reading will in guide scaleportion N For example, find the square root of 25. The number is in thescale portion N and is of the form 10m. Therefore, the L scale readingcorresponding tothe scale portion N setting of 25, namely, 3979 is setin section I of the L/2 scale, and the square root 5 is found in thelower guide scale portion N under the cross-hair.

Cube root. Set invthe guide scale portion N or N the cross-hair of thecursor to the number of which the cube root is desired. Take the L scalereading. Note whether the number is in the scale portion N, or N andwhether it has the form of m or 10m or 100m. Under these two types ofheadings for cube roots in the above table will be found a Roman numeralaccompanying bl or N The Roman numeral I, II, or III designates thesection of the L/3 scale in which the L scale reading is to be set.Having made this setting the said N or N' indicate the guide scaleportion in which the cube root of the number is to be found.

"Rules applying only to the special case embodied in this invention areas follows: If the number is of the form In the L scale reading will beset in section I or II of the L/il scale according as the number occursin the guide scale portion N or N If the number is of the form 10m the Lscale reading will he set in section III or I of the L/3 scale accordingas the number occurs in guide scale port-ion N or N If the number is ofthe form 100m the L scale reading will be set in section'II orIII of theL/3 scale according as the number occurs in guide scale portion N or NIf the L scale setting is made in section I or III of the L/3 scale thecube root is foundin the same scale portion N or N as the number. If theL scale setting is made in section II of the L/3 scale the root is foundin guide scale ortion N, or N according as the number is in guidescaleportion N or N,, respectively.

For example, find the cube root of 27. The number is in scale portion Nand is of the form 10m. Therefore, the L scale setting corresponding tothe scale portion N setting of 27, namely, 4314 is to be made in sectionIII of the L/3 scale and the cube root 3 is found in the upper guidescale portion N under the cross-hair.

Eoolutiom- Square of a number. Set the cross-hair of the cursor to thenumber in guide scale portion N or N,. Take the L/2' Another ruleapplying only to the special case embodied in this invention is asfollows. The square of a number is found in guide scale portion N or Naccording as the L/2 scale reading is made in section I or II,respectively.

For example, find the square of 2. The L/2 scale reading 6021corresponding to the upper guide scale setting of 2 is in section II.Therefore, the 6021 setting made in the L scale gives the square 4 inthe lower guide scale portion N under the cross-hair.

Cube of a number. Set the cross-hair of the cursor to the number inguide scale rtion N or N,. Take the reading in the /3 scale and notewhether the section is I, II, or III in which the reading is taken. Setthe L/3 scale reading in the L scale. The above Roman numeralaccompanied with N or N,, in whichever scale portion the number occurs,gives in the above table the scale portion N or N in which the cube isfound. Rules applying only to the special case embodied in thisinvention are as follows. The cube of a number is found in the sameguide scale portion N or N as the number if the L/ 3 scale reading istaken in section I or III. The cube is found in the other scale portionif the L/3 reading is taken in section II.

For example, find the cube of 2. The L/3 scale reading 9031 corresonding to t? 9 upper guidescale setting 0 2 is .in section II. Therefore, the setting 9031 made in the L scale gives the cube 8 in the lowerguide scale portion N, under the cross-hair.

It is evident that the special rules regarding square roots, cube roots,squares, and cubes may be indicated on the L, L/2, and L/3 scales sothat the cumbersome table may be omitted. Furthermore, the use of alogarithmic scale divided into three 1 qual linear portions placed onebelow another in registry accompanying scales L and L/ 3 render simplerules for cubes and cube roots similar to the rules for squares and suare roots applying to the use of a logarit mic scale divided into twoequal linear portions accompanying scales L and L/2.

Logarithm of a nmnber.Set in the guide scale portion N or N thecross-hair of the cursor to the number of which the logarithm is wanted.If the number is in the up er guide scale portionN the L scale readinggives the mantissa of the lo arithm directly, the decimal point beingaced to the left of the left hand digit. I?

be added to the L scale reading properly ointed off. The usual rules forcharacter istic must be observed.

For example find the logarithm of 4. The L scale rea ing is .1021. Thenumber is in the lower guide scale ortion N,. The characteristic of thenum er is 0. Therefore, the logarithm is 0.6021.

I claim:

In a logarithmic calculatin device having two members capable oiparallel rethe number is inthe lower guide scale portion N 0.5 must

